Transform flat surface into paraboloid

Question

Is it possible to transform a flat surface into a paraboloid

$$z=x^2+y^2$$

such that there is no strain in the circular in the circular cross section (direction vector A)?

If the answer is yes, is it possible to calculate the shape of such a flat surface?

Where can I find more information to solve this kind of problems?

Answer 1

Since it seems my solution is what the OP wanted...

Consider the parametric equations

$$\begin{align*} x&=cu\cos\,\theta\\ y&=cu\sin\,\theta\\ z&=h(1-u^2) \end{align*}$$

with the parameter ranges $0\leq u\leq 1$ and $0\leq\theta\leq2\pi$.

For $h=0$, you have a disk of radius $c$; from here, varying $h$ in either the positive or negative directions will yield a paraboloid of revolution:

Answer 2

If I understand correctly your phrase "no strain in the circular cross section," the answer to your question is No. The paraboloid is not a developable surface. A developable surface has zero Gaussian curvature everywhere, whereas the paraboloid has positive curvature. The analogy here is to a sphere, also not developable, which is why there is such a plethora of map projections aiming for minimal visual distortion of a non-developable surface.

Update. See the comments. I did not interpret the strain condition as the OP intended, apparently. The comments clarify.